Problem 24 - Advanced Engineering Mathematics

Question

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{ccc} 1 & 1 & 1 \\ x & y & z \\ 2+x & 3+y & 4+z \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant is \(2y - x - z\).

Step 1: Choose the Row or Column

In this problem, we can choose any row or column to perform a cofactor expansion. Let's choose the first row for simplicity. The elements of the first row are 1, 1, and 1.

Step 2: Compute the Cofactors

The cofactor of an element is calculated as \((-1)^{i+j}\) times the determinant of the minor formed by removing the corresponding row and column. - For the element 1 in the first row and first column, the minor is \(\begin{vmatrix} y & z \ 3+y & 4+z \end{vmatrix}\).- For the element 1 in the first row and second column, the minor is \(\begin{vmatrix} x & z \ 2+x & 4+z \end{vmatrix}\).- For the element 1 in the first row and third column, the minor is \(\begin{vmatrix} x & y \ 2+x & 3+y \end{vmatrix}\).

Step 3: Calculate the Determinants of 2x2 Minors

Compute the determinant of each 2x2 minor:- \(\begin{vmatrix} y & z \ 3+y & 4+z \end{vmatrix} = y(4+z) - z(3+y) = 4y + yz - 3z - yz = 4y - 3z\).- \(\begin{vmatrix} x & z \ 2+x & 4+z \end{vmatrix} = x(4+z) - z(2+x) = 4x + xz - 2z - xz = 4x - 2z\).- \(\begin{vmatrix} x & y \ 2+x & 3+y \end{vmatrix} = x(3+y) - y(2+x) = 3x + xy - 2y - xy = 3x - 2y\).

Step 4: Apply the Cofactor Signs

Apply the cofactor signs for the row expansion (first row):- For the first element (1, cofactor sign is \((-)^{1+1} = +1\)), the cofactor is \(4y - 3z\).- For the second element (1, cofactor sign is \((-)^{1+2} = -1\)), the cofactor is \(-1(4x - 2z) = -4x + 2z\).- For the third element (1, cofactor sign is \((-)^{1+3} = +1\)), the cofactor is \(3x - 2y\).

Step 5: Perform the Cofactor Expansion

Add up the products of each element of the first row and its corresponding cofactor to evaluate the determinant. The determinant is: \[1(4y - 3z) + 1(-4x + 2z) + 1(3x - 2y) = 4y - 3z - 4x + 2z + 3x - 2y\].

Step 6: Simplify the Expression

Combine like terms in the expansion:\[4y - 2y - 4x + 3x - 3z + 2z = 2y - x - z\].Therefore, the determinant is \(2y - x - z\).

Key Concepts

Cofactor Expansion

Cofactor expansion is a method used in matrix algebra to calculate the determinant of a square matrix. It is a handy tool, especially for larger matrices, as it breaks down the original matrix into smaller and more manageable pieces. To perform a cofactor expansion:

• Choose a row or a column of the matrix. In practice, always select the one with the most zeros for simplicity.
• For each element in the chosen row or column, calculate the minor. This involves removing the row and column of the element to form a new, smaller matrix.
• Determine the cofactor, which is the minor's determinant multiplied by \((-1)^{i+j}\), where \i\ and \j\ are the row and column indices of the element.

Once cofactors are calculated, multiply each element of the chosen row or column by its corresponding cofactor. Finally, sum these products to get the determinant of the matrix. Understanding cofactor expansion is crucial because it lays the foundation for understanding more complex operations in matrix algebra.

Matrix Algebra

Matrix algebra involves operations and methods used to manipulate matrices in order to solve systems of linear equations, among other applications. Key operations in matrix algebra include addition, multiplication, and finding determinants.

Determinants are particularly important because they tell us if a matrix is invertible. Only matrices with non-zero determinants have inverses. In the context of solving systems, if the determinant of the coefficient matrix is zero, the system is either dependent or inconsistent.

Matrix algebra not only focuses on operations but also on understanding the properties of matrices. For instance:

• Associative property (A(BC) = (AB)C) for multiplication
• Additive identity, where adding a zero matrix to any matrix A gives A itself
• Multiplicative identity, where multiplying a matrix by the identity matrix leaves it unchanged

These properties are integral to grasping how matrices function as mathematical tools.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, and linear mappings between these spaces. It extends its utility beyond just solving algebraic equations, touching upon various fields including computer science, engineering, and physics.

One fundamental aspect of linear algebra is its ability to solve systems of linear equations through matrix representations. By using matrices, we can achieve solutions more efficiently compared to traditional algebraic methods. Linear transformations, a pivotal concept in linear algebra, help describe linear mappings of spaces, allowing us to understand changes and movements within different dimensions.

Moreover, vectors and matrices in linear algebra encapsulate more complex forms of data, enabling broader applications. For instance:

• Vectors can represent points, directions, or layers within multi-dimensional spaces
• Matrices can be seen as linear transformations acting on vector spaces
• Eigenvalues and eigenvectors, which convey how and in which direction transformations occur

Grasping linear algebra concepts is central to delving deeper into advanced mathematical topics and real-world applications.